**Abstract:** In 2003, Prescott hypothesized a special sorting operation performed on the genomic material of ciliates. This operation, called cds, involves block interchanges of permuted lists. Christie (1996) discovered that among those permutations which are sortable by cds, cds sorts them using the fewest possible block interchanges of any kind. Adamyk et al. (2013) discovered an efficient way of quantifying the non-cds-sortability of a permutation called the strategic pile. My group partially characterized permutations with maximal strategic pile (in some sense, the permutations for which applying cds is the most unpredictable), completing this characterization when the number of available cds moves is minimal and when it is nearly maximal. We discover a ZnｘZn-action on permutations in S(n+1) that preserves the number of available cds moves. This group action, defined on permutations, has a very nice visual interpretation via chord diagrams (circles with intersecting chords). We characterize the stabilizer of this group action, and are thus able to use Lagrange’s Theorem to count the size of orbits of permutations. This allows us to count the number of permutations with a given number of cds moves.